Reteach. Congruence and Transformations

Transcription

1 Congruence and Transformations TYPES OF TRANSFORMATIONS (centered at (0, 0)) Translation (slide): (x, y) (x a, y b) Reflection y-axis: (x, y) ( x, y) x-axis: (x, y) (x, y) Rotation 90 clockwise: (x, y) (y, x) Dilation: (x, y) (kx, ky), k 0 Rotation 90 counterclockwise: (x, y) ( y, x) Rotation 180 : (x, y) ( x, y) Apply the transformation M to the polygon with the given vertices. Identify and describe the transformation. 1. M: (x, y) (x 1, y 2) A( 1, 3), B(2, 2), C( 2, 1) 2. M: (x, y) ( x, y) P(0, 0), Q(1, 3), R(3, 3)

2 Congruence and Transformations continued An isometry is a transformation that preserves length, angle, and area. Because of these properties, isometries produce congruent images. A rigid transformation is another name for an isometry. Dilations with scale factor k 1 are transformations that produce images that are not congruent to their preimages. Transformation Isometry Image Preimage translation yes yes reflection yes yes rotation yes yes dilation no no Determine whether the polygons with the given vertices are congruent. 3. E( 3, 1), F( 2, 4), G(0, 0) H(1, 4), I(2, 1), J(4, 5) 4. R( 2, 4), S(0, 3), T( 3, 1) U(2, 4), S(0, 3), V(3, 1) 5. P(0, 0), Q(2, 2), R( 2, 1) P(0, 0), S(4, 4), T( 4, 2) 6. J( 2, 2), K(2, 1), L(1, 3) P( 4, 4), Q(4, 2), R(2, 6)

3 Angle Relationships in Triangles According to the Triangle Sum Theorem, the sum of the angle measures of a triangle is m J m K m L The corollary below follows directly from the Triangle Sum Theorem. Corollary The acute angles of a right triangle are complementary. m C Example m C m E 908 Use the figure for Exercises 1 and Find m ABC. 2. Find m CAD. Use RST for Exercises 3 and What is the value of x? 4. What is the measure of each angle? What is the measure of each angle? 5. L 6. C 7. W

4 Angle Relationships in Triangles continued An exterior angle of a triangle is formed by one side of the triangle and the extension of an adjacent side. 1 and 2 are the remote interior angles of 4 because they are not adjacent to 4. Exterior Angle Theorem The measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles. m 4 m 1 1 m 2 Third Angles Theorem If two angles of one triangle are congruent to two angles of another triangle, then the third pair of angles are congruent. Find each angle measure. 8. m G 9. m D Find each angle measure. 10. m M and m Q 11. m T and m R

5 Name Date Class Congruent Triangles Triangles are congruent if they have the same size and shape. Their corresponding parts, the angles and sides that are in the same positions, are congruent. Corresponding Parts Congruent Angles A J B K C L Congruent Sides AB JK BC KL CA LJ To identify corresponding parts of congruent triangles, look at the order of the vertices in the congruence statement such as ABC JKL. Given: XYZ NPQ. Identify the congruent corresponding parts. 1. Z 2. YZ 3. P 4. X 5. NQ 6. PN Given: EFG RST. Find each value below. 7. x 8. y 9. m F 10. ST

6 Congruent Triangles continued You can prove triangles congruent by using the definition of congruence. Given: D and B are right angles. DCE BCA C is the midpoint of DB. ED AB, EC AC Prove: EDC ABC Proof: Statements Reasons 1. D and B are rt. s. 1. Given 2. D B 2. Rt.< Thm. 3. DCE BCA 3. Given 4. E A 4. Third s Thm. 5. C is the midpoint of DB. 5. Given 6. DC BC 6. Def. of mdpt. 7. ED AB, EC AC 7. Given 8. EDC ABC 8. Def. of s 11. Complete the proof. Given: Q R P is the midpoint of QR. NQ SR, NP SP Prove: NPQ SPR Statements Reasons 1. Q R 1. Given 2. NPQ SPR 2. a. 3. N S 3. b. 4. P is the midpoint of QR. 4. c. 5. d. 5. Def. of mdpt. 6. NQ SR, NP SP 6. e. 7. Houghton NPQ Mifflin Harcourt SPR Publishing Company 7. f.

7 Triangle Congruence: SSS and SAS Side-Side-Side (SSS) Congruence Postulate If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent. QR TU, RP US, and PQ ST, so PQR STU. You can use SSS to explain why FJH > FGH. It is given that FJ JH GH. By the Reflex. FG and that Prop. of, FH FH. So FJH FGH by SSS. Side-Angle-Side (SAS) Congruence Postulate If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. Use SSS to explain why the triangles in each pair are congruent. 1. JKM LKM 2. ABC CDA 3.Use SAS to explain why WXY WZY.

8 You can show that two triangles are congruent by using SSS and SAS Show that JKL FGH for y 7 HG y 6 m G 5y 5 FG 4y (7) (7) 1 27 HG LK 13, so HG LK by def. of segs. m G 40, so G K by def. of FG JK 27, so FG JK by def. of segs. Therefore JKL FGH by SAS. Show that the triangles are congruent for the given value of the variable.

9 Isosceles and Equilateral Triangles Theorem Examples Isosceles Triangle Theorem If two sides of a triangle are congruent, then the angles opposite the sides are congruent. If RT RS, then <T S. Converse of Isosceles Triangle Theorem If two angles of a triangle are congruent, then the sides opposite those angles are congruent. If <N M, then LN LM. You can use these theorems to find angle measures in isosceles triangles. Find m E in DEF. m D m E 5x (3x 14) given values. 2x 14 both sides. x 7 sides by 2. Thus m E 3(7) Isosc. Thm. Substitute the Subtract 3x from Divide both Find each angle measure.

10 Isosceles and Equilateral Triangles continued Equilateral Triangle Corollary If a triangle is equilateral, then it is equiangular. (equilateral equiangular ) Equiangular Triangle Corollary If a triangle is equiangular, then it is equilateral. (equiangular equilateral ) If <A B C, then AB BC CA.

11 Properties of Parallelograms A parallelogram is a quadrilateral with two pairs of parallel sides. All parallelograms, such as FGHJ, have the following properties. Properties of Parallelograms FG HJ GH JF Opposite sides are congruent. m F m G 180 m G m H 180 m H m J 180 m J m F 180 Consecutive angles are supplementary. F H G J Opposite angles are congruent. FP HP GP JP The diagonals bisect each other.

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